Transformations

1
Chapter 7

• Transformations

2
Chapter Objectives

• Identify different types of transformations
• Define isometry
• Identify reflection
• Identify rotations
• Identify translations
• Describe composition transformations

3
Lesson 7.1

• Rigid Motion in a Plane

4
Lesson 7.1 Objectives

• Identify basic rigid transformations
• Define isometry

5
Definition of Transformation

• A transformation is any operation that maps, or
  moves, an object to another location or
  orientation.

6
Transformation Terms

• When performing a transformation, the original
  figure is called the pre-image.
• The new figure is called the image.
• Many transformations involve labels
• The image is named after the pre-image, by adding
  a prime symbol (apostrophe)
• A A A
• We say it as A prime

7
Types of Transformations
Types Reflection Rotation Translation
Characteristics
Orientation
Pictures
Flips object over line of reflection
Turns object using a fixed point as center or
rotation
Slides object through a plane
Stays same just tilted
Order in which object is drawn is reversed
Stays same and stays upright
8
Definition of Isometry

• An isometry is a transformation that preserves
  length.
• Isometry also preserve angle measures, parallel
  lines, and distances between points.
• If you look at the meaning of the two parts of
  the word, iso- means same, and metry- means meter
  or measure.
• So simply stated, isometry preserves size.
• Any transformation that is an isometry is called
  a Rigid Transformation.

9
Homework 7.1

• 1-33, 36-39
• p399-401
• In Class 9, 13, 27, 33
• Due Tomorrow

10
Lesson 7.2

• Reflections

11
Lesson 7.2 Objectives

• Utilize reflections in a plane
• Define line symmetry
• Derive formulas for specific reflections in the
  plane

12
Reflections

• A transformation that uses a line like a mirror
  is called a reflection.
• The line that acts like a mirror is called the
  line of reflection.
• When you talk of a reflection, you must include
  your line of reflection
• A reflection in a line m is a transformation that
  maps every point P in the plane to a point P, so
  that the following is true
• If P is not on line m, then m is the
  perpendicular bisector of PP.
• If P is on line m, then PP.

13
Theorem 7.1Reflection Theorem

• A reflection is an isometry.
• That means a reflection does not change the shape
  or size of an object!

14
Line of Symmetry

• A figure in the plane has a line of symmetry if
  the figure can be mapped onto itself by a
  reflection in a line.
• What that means is a line can be drawn through an
  object so that each side reflects onto itself.
• There can be more than one line of symmetry, in
  fact a circle has infinitely many around.

15
Homework 7.2a

• 1-11, 22-29
• p407-408
• In Class 7, 23
• Due Tomorrow

16
Reflection Formula

• There is a formula to all reflections.
• It depends on which type of a line are you
  reflecting in.
• vertical
• horizontal
• y x

Vertical y-axis x a
Horizontal x-axis y a
y x
( x , y)
( -x 2a , y)
( x , -y 2a)
( y , x)
17
Homework 7.2b

• 12-14, 18-21, 50-51
• p407-410
• In Class 19
• Due Tomorrow

18
Lesson 7.3

• Rotations

19
Lesson 7.3 Objectives

• Utilize a rotation in a plane
• Define rotational symmetry
• Observe any patterns for rotations about the
  origin

20
Definitions of Rotations

• A rotation is a transformation in which a figure
  is turned about a fixed point.
• The fixed point is called the center of rotation.
• The amount that the object is turned is the angle
  of rotation.
• A clockwise rotation will have a negative
  measurement.
• A counterclockwise rotation will have a positive
  measurement.

clockwise or negative (-)
21
Theorem 7.2Rotation Theorem

• A rotation is an isometry.

22
Rotational Symmetry

• A figure in a plane has rotational symmetry if
  the figure can be mapped onto itself by a
  rotation of 180 or less.
• A square has rotational symmetry because it maps
  onto itself with a 90 rotation, which is less
  than 180.
• A rectangle has rotational symmetry because it
  maps onto itself with a 180 rotation.

23
Homework 7.3a

• 1-19
• p416
• In Class 6, 11, 13
• Due Tomorrow

24
Rotating About the Origin

• Rotating about the origin in 90o turns is like
  reflecting in the line y x and in an axis at
  the same time!
• So that means to switch the positions of x and y.
• (x,y) ? (y,x)
• Then the original x-value changes sign, no matter
  where it is flipped to.
• So overall the transformation can be described by
• (x,y) ? (-y,x)
• Every time you 90o you repeat the process.
• So going 180o means you do the process twice!

25
Theorem 7.3Angle of Rotation Theorem

• The angle of rotation is twice as big as the
  angle of intersection.
• But the intersection must be the center of
  rotation.
• And the angle of intersection must be acute or
  right only.

26
Homework 7.3b

• 25-35, 45-50, 54
• p417-419
• In Class 25, 35
• Due Tomorrow
• Quiz Wednesday
• Lessons 7.1-7.4

27
Lesson 7.4

• Translations
• and
• Vectors

28
Lesson 7.4

• Define a translation
• Identify a translation in a plane
• Use vectors to describe a translation
• Identify vector notation

29
Translation Definition

• A translation is a transformation that maps an
  object by shifting or sliding the object and all
  of its parts in a straight light.
• A translation must also move the entire object
  the same distance.

30
Theorem 7.4Translation Theorem

• A translation is an isometry.

31
Theorem 7.5Distance of Translation Theorem

• The distance of the translation is twice the
  distance between the reflecting lines.

x
2x
32
Coordinate form

• Every translation has a horizontal movement and a
  vertical movement.
• A translation can be described in coordinate
  notation.
• (x,y) ? (xa , yb)
• Which tells you to move a units horizontal and b
  units vertical.

Q
b units up
P
a units to the right
33
Vectors

• Another way to describe a translation is to use a
  vector.
• A vector is a quantity that shows both direction
  and magnitude, or size.
• It is represented by an arrow pointing from
  pre-image to image.
• The starting point at the pre-image is called the
  initial point.
• The ending point at the image is called the
  terminal point.

34
Component Form of Vectors

• Component form of a vector is a way of combining
  the individual movements of a vector into a more
  simple form.
• ltx , ygt
• Naming a vector is the same as naming a ray.
• PQ

Q
y units up
P
x units to the right
35
Use of Vectors

• Adding/subtracting vectors
• Add/subtract x values and then add y values
• lt2 , 6gt lt3 , -4gt
• lt5 , 2gt
• Distributive property of vectors
• Multiply each component by the constant
• 5lt3 , -4gt
• lt15 , -20gt
• Length of vector
• Pythagorean Theorem
• x2 y2 lenght2
• Direction of vector
• Inverse tangent
• tan-1 (y/x)

36
Homework 7.4

• 1-30, 44-47
• p425-427
• In Class 3,7,17,25,45
• Due Tomorrow
• Quiz Tomorrow
• Lessons 7.1-7.4

37
Lesson 7.5

• Glide Reflections
• and
• Compositions

38
Lesson 7.5 Objectives

• Identify a glide reflection in a plane
• Represent transformations as compositions of
  simpler transformations

39
Glide Reflection Definition

• A glide reflection is a transformation in which a
  reflection and a translation are performed one
  after another.
• The translation must be parallel to the line of
  reflection.
• As long as this is true, then the order in which
  the transformation is performed does not matter!

40
Compositions of Transformations

• When two or more transformations are combined to
  produce a single transformation, the result is
  called a composition.
• So a glide reflection is a composition.
• The order of compositions is important!
• A rotation 90o CCW followed by a reflection in
  the y-axis yields a different result when
  performed in a different order.

41
Theorem 7.6Composition Theorem

• The composition of two (or more) isometries is an
  isometry.

42
Homework 7.5

• 1-8, 9-21, 23-24, 26-30
• skip 16, 28
• p433-435
• In Class 9,13,19
• Due Tomorrow

43
Lesson 7.6

• Frieze Patterns

44
Lesson 7.6 Objectives

• Identify a frieze pattern by type
• Visualize the different compositions of
  transformations

45
Frieze Patterns

• A frieze pattern is a pattern that extends to the
  left or right in such a way that the pattern can
  be mapped onto itself by a horizontal
  translation.
• Also called a border pattern.

46
Classifying Frieze Patterns

• The horizontal translation is the minimum that
  must exist.
• However, there are other transformations that can
  occur.
• And they can occur more than once.

Type Abbreviation Description





Translation
Horizontal translation left or right
T
180o Rotation
180o Rotation CW or CCW
R
Reflection either up or down in a horizontal line
Reflection in Horizontal Line
H
Reflection in Vertical Line
Reflection either left or right in a vertical line
V
Horizontal translation with reflection in a
horizontal line
Horizontal Glide Reflection
G
47
Examples
TR
TG
TV
THG
TRVG
TRHVG
48
Homework 7.6

• 2-23
• p440-441
• In Class 9,13,17,21
• Due Tomorrow
• Quiz Tuesday
• Lessons 7.5-7.6